A random sample of n 64 observations is drawn from a popula

A random sample of n = 64 observations is drawn from a population with

mean equal to 20 and standard deviation equal to 16.

(a) What is the mean and standard deviation for the sampling distribution of x?

(b) Find P(16 < x < 23).

Solution

a)

It will have the same mean,

ux = 20 [answer]

However, the standard deviation will be divided by the square root of n,

sx = 16/sqrt(64) = 2 [answer]

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b)

Do you mean x here as in the sample mean (xbar)? If so:

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    16      
x2 = upper bound =    23      
u = mean =    20      
          
s = standard deviation =    2      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -2      
z2 = upper z score = (x2 - u) / s =    1.5      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.022750132      
P(z < z2) =    0.933192799      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.910442667   [ANSWER]

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If you meant them as individual scores,

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    16      
x2 = upper bound =    23      
u = mean =    20      
          
s = standard deviation =    8      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -0.5      
z2 = upper z score = (x2 - u) / s =    0.375      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.308537539      
P(z < z2) =    0.646169767      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.337632228   [answer]